Two software companies are developing new android apps. The average profit per x hundred downloads earned by each company can be modelled by the rational functions below, where x > or equal to 0.
Trance Media: G(x)=80x+150/x
Ellen's incorporated: B(x)=35x+600/x
How many downloads are required for the profit earned by Trance Media to be grater than the profit earned by Elena incorporated? Provide full algebraical solution, including a sign chart.
We want
80x+150/x ≥ 35x+600/x
45x ≥ 450/x
times x , (since clearly x is positive, the inequality sign does not change)
45x^2 ≥ 450
x^2 ≥ 10
x ≥ √10 or appr 3.162 thousand downloads, that is 3162 downloads
Thank you so much!!!!
what would be the intervals i would use to make the chart?
To find the number of downloads required for the profit earned by Trance Media to be greater than the profit earned by Ellen's Incorporated, we need to compare the two rational functions and determine the point at which G(x) > B(x).
The rational functions given are:
G(x) = 80x + 150/x
B(x) = 35x + 600/x
Step 1: Set up the inequality
We want to find when G(x) is greater than B(x), so we set up the inequality:
G(x) > B(x)
Step 2: Substitute the rational functions
Substitute the given rational functions into the inequality:
80x + 150/x > 35x + 600/x
Step 3: Combine like terms
Combine the terms on each side of the inequality:
80x - 35x > 600/x - 150/x
Simplifying, we get:
45x > 450/x
Step 4: Multiply through by x
To remove the denominators, multiply both sides of the inequality by x:
45x^2 > 450
Step 5: Subtract 450 from both sides
Subtract 450 from both sides of the inequality:
45x^2 - 450 > 0
Step 6: Simplify the inequality
Divide both sides of the inequality by 45 to simplify:
x^2 - 10 > 0
Step 7: Create a sign chart
To find the solution to the inequality, create a sign chart by selecting test points and determining the sign of the expression x^2 - 10 for each interval.
Test point 1: x = 0
Plugging in x = 0 gives:
(0)^2 - 10 = -10
The expression is negative in this interval.
Test point 2: x = 1
Plugging in x = 1 gives:
(1)^2 - 10 = -9
The expression is negative in this interval.
Test point 3: x = 2
Plugging in x = 2 gives:
(2)^2 - 10 = -6
The expression is negative in this interval.
Test point 4: x = 5
Plugging in x = 5 gives:
(5)^2 - 10 = 15
The expression is positive in this interval.
Step 8: Determine the solution
From the sign chart, we can see that the expression x^2 - 10 is positive for x > 5. This means that for G(x) to be greater than B(x), the number of downloads (x) must be greater than 5.
Therefore, greater than 5 hundred downloads are required for the profit earned by Trance Media to be greater than the profit earned by Ellen's Incorporated.